I have obtained two data sets which I have already proven to have an inverse relationship. However, can I used coefficient of determination (R^2) to determine the correlation between the two variables? I'm not sure if it can be used for other relationships other than linear. Thanks!
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If you have a model (linear or not) $y=f(x)$ then $R^2$ will be a measure of how well your model captures the variance of $y$.
If you want a model-free ways to measure the strength of a non-linear relationship, I'd suggest Spearman's Rank Correlation instead of Pearson's Moment Correlation. Spearman's rank will give intuitive answers for any monotonic relationship (linear or not).
Annika
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